Subsets of 1 to 10 with a single adjacent pair

Count the subsets of $\{1,2,\dots,10\}$ in which exactly one pair of elements are consecutive integers. For instance, $\{3,5,6,9\}$ qualifies because $5,6$ is its only consecutive pair.

$235$. Glue the consecutive pair into one block, so the set is $j$ isolated singletons plus that block, with gaps between all pieces. Placing these $j+1$ pieces in $10$ slots can be done in $\binom{9-j}{j+1}$ ways, and any piece may be the block, so the count is $\sum_{j}(j+1)\binom{9-j}{j+1}=9+56+105+60+5=235$.